1 / 11

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 12, Friday, September 26

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 12, Friday, September 26. Homework. Homework (MATH 310#4F): Read 4.1. Do Exercises 3.2: 6,14,17,19,20 Volunteers: ____________ ____________ Problem: 19 . Homework #4 is due Wednesday. Theorem 3.

olsonk
Download Presentation

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 12, Friday, September 26

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MATH 310, FALL 2003(Combinatorial Problem Solving)Lecture 12, Friday, September 26

  2. Homework • Homework (MATH 310#4F): • Read 4.1. • Do Exercises 3.2: 6,14,17,19,20 • Volunteers: • ____________ • ____________ • Problem: 19. • Homework #4 is due Wednesday.

  3. Theorem 3 • Let T be an m-ary tree of height h with l leaves. Then • (a) l · mh, and if all leaves are at height h, l = mh. • (b) h ¸d logml e, and if the tree is balanced, • h = d logml e.

  4. Example 4 • There is a balance and there are n = 8 coins, one of which is counterfeit (too light). • How many weighings are needed? 123|678 1|3 4|5 6|8 1 2 3 4 ? 5 6 7 8

  5. A Spanning Tree • Each connected graph has a spanning tree. • For finite graphs the proof is easy. [Keep removing edges that belong to some circuit]. • A graph has a spanning tree if and only if it is connected.

  6. Spanning Trees – More New Terms • More new terms: • depth-first spanning tree • breadth-first spanning tree • adjacency matrix

  7. Example 1. Testing For Connectedness • Is the undirected graph whose adjacency matrix is given on the left connected?

  8. Example 2: Traversing a Maze • Consider the maze in Figure 3.8. on p.108. • We start a location marked with an S and seek to reach the end marked with an E. • Depth-First Search • Breadth-First Search

  9. Example 3: Pitcher Pouring Puzzle • We are given three pitchers of water, of sizes 10 quarts, 7 quarts and 4 quarts. • Initially, the 10-quart is full and the other two pitchers are empty. • Is there a way to pour among pitchers to obtain exactly 2 quarts in one of the pitchers?

  10. Example 4: Jealous Wives Puzzle. • Three jealous wives and three husbands come to a river. • The boat is for two people. • Find a sequence of boat trips that will get 6 people across the river without ever letting any husband to be without his wife in the presence of another wife.

  11. A Center of a Tree T • Let T be an arbitrary tree and let r be any of its vertices. Let Tr be the tree rooted at r. Vertex r is called a center if the correspondig rooted tree Tr has minimal height. Compare Exercise 18, p. 103. 7 3 2 4 8 1 6 5

More Related